Nonlinear dimensionality reduction for nonadiabatic dynamics: Theinfluence of conical intersection topography on population transfer ratesAaron M. Virshup, Jiahao Chen, and Todd J. Martínez Citation: J. Chem. Phys. 137, 22A519 (2012); doi: 10.1063/1.4742066 View online: http://dx.doi.org/10.1063/1.4742066 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v137/i22 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 137, 22A519 (2012)

Nonlinear dimensionality reduction for nonadiabatic dynamics: Theinfluence of conical intersection topography on population transfer rates

Aaron M. Virshup,1 Jiahao Chen,2 and Todd J. Martínez3,4

1Department of Chemistry, Duke University, Durham, North Carolina 27710, USA2Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA3Department of Chemistry and The PULSE Institute, Stanford University, Stanford, California 94305, USA4SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA

(Received 20 May 2012; accepted 19 July 2012; published online 9 August 2012)

Conical intersections play a critical role in the nonadiabatic relaxation of excited electronic states.However, there are an infinite number of these intersections and it is difficult to predict which are ac-tually relevant. Furthermore, traditional descriptors such as intrinsic reaction coordinates and steepestdescent paths often fail to adequately characterize excited state reactions due to their highly nonequi-librium nature. To address these deficiencies in the characterization of excited state mechanisms, weapply a nonlinear dimensionality reduction scheme (diffusion mapping) to generate reaction coor-dinates directly from ab initio multiple spawning dynamics calculations. As illustrated with variousexamples of photoisomerization dynamics, excited state reaction pathways can be derived directlyfrom simulation data without any a priori specification of relevant coordinates. Furthermore, diffu-sion maps also reveal the influence of intersection topography on the efficiency of electronic popu-lation transfer, providing further evidence that peaked intersections promote nonadiabatic transitionsmore effectively than sloped intersections. Our results demonstrate the usefulness of nonlinear di-mensionality reduction techniques as powerful tools for elucidating reaction mechanisms beyondthe statistical description of processes on ground state potential energy surfaces. © 2012 AmericanInstitute of Physics. [http://dx.doi.org/10.1063/1.4742066]

I. INTRODUCTION

The faithful modeling of excited state reaction dynamicsis a major challenge since it places great demands on bothelectronic structure theory (the calculation of electronic ex-cited states for highly nonequilibrium molecular geometries)and dynamics (quantum effects must be included to describethe nonadiabatic transitions which allow for the transfer ofenergy from the electronic to nuclear degrees of freedom).Ab initio molecular dynamics (AIMD) methods, wherethe dynamics and electronic structure problems are solvedsimultaneously, have been developed for nonadiabaticdynamics1–10 in order to avoid the need to fit potential energysurfaces and their couplings to analytic functional forms. Anoft-quoted advantage of AIMD methods is that all nuclear de-grees of freedom can be included in the dynamics, i.e., there islittle or no incentive to create reduced dimensionality models.Ironically, while this is indeed an advantage for the realisticmodeling of excited state dynamics, it can be an obstacle toextracting chemical understanding. Simply put, as more de-grees of freedom (electronic and/or vibrational) are included,it can become harder to see which are the most important.Thus, a means of automatically identifying the important de-grees of freedom would be most welcome.

For chemical processes occurring entirely on the groundelectronic state, one can use statistical theories as a guide toidentify important degrees of freedom. Thus, one identifieslocal minima and transition states connecting these minimaas the first step in this process. One can further find min-imal energy pathways connecting these points and thereby

identify possible reactions and their mechanisms. This ap-proach is not so straightforward for many excited state re-actions because they are intrinsically far from equilibrium.Excited state reaction dynamics typically take place on fem-tosecond timescales, as the electronic excitation event is rapidand the molecule often finds itself on a steeply sloped partof the excited state potential energy surface after photon ab-sorption. The nonequilibrium nature of photodynamics thuspresents some difficulty in establishing quantitative rate theo-ries. The standard methods for determining reaction paths andrates for ground state reactions rely primarily on the assump-tions of transition state theory; in particular, the dynamicalevolution of the system is either neglected entirely or at besttreated only in a thermodynamically averaged sense. Reac-tion coordinates arising from such assumptions, such as theone-dimensional intrinsic reaction coordinate (IRC) that pa-rameterizes the minimal energy pathway,11 are therefore un-likely to provide a realistic description of ultrafast processeswhich are completed in picoseconds or even femtoseconds.This problem is not unique to excited state reactions: dynam-ical correlations are already known to disrupt the minimal en-ergy path picture even in some thermally activated groundstate reactions.12–14 While transition path sampling methodsbased on dynamical trajectories have been developed withgreat success as a means of correctly sampling a variety ofcandidate reaction pathways,15 they have so far only been ex-plored for reactions on the ground electronic state.

Conical intersections (CIs), i.e., molecular geometrieswhere two or more electronic states are exactly degenerate,

0021-9606/2012/137(22)/22A519/10/$30.00 © 2012 American Institute of Physics137, 22A519-1

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22A519-2 Virshup, Chen, and Martínez J. Chem. Phys. 137, 22A519 (2012)

are now recognized to play a central role in ultrafastphotochemistry.16, 17 The Born-Oppenheimer approximationbreaks down at and near these geometries, thus allowingnonadiabatic population transfer between different electronicstates. Thus, these geometries may be viewed as analogs tothe transition state in a ground state chemical reaction (buthere for the “reaction” A* → A). However, CIs are not iso-lated points, but rather high-dimensional seams. For a CI in-volving two electronic states, the degeneracy between the po-tential energy surfaces (PESs) is lifted linearly in exactly twodimensions (the “branching plane”) around the CI, formingthe shape of a double cone. To first order, the two electronicstates remain degenerate with respect to molecular displace-ments along all other vibrational degrees of freedom. Thus,for a two state intersection in a molecule with N vibrationaldegrees of freedom, the CI seam is an N-2 dimensional mani-fold. It can be difficult to predict which parts of this manifoldwill be most important in a given excited state reaction.

In this work, we aim to construct coarse-grained rep-resentations of dynamically accessed reaction paths andportions of the CI seam directly from dynamics simulations.One goal is to identify the dynamically accessible seam ofCIs. In general, the shape of the CI manifold is difficultto determine, although in some cases, certain points onthe CI manifold can be deduced from simple symmetryconsiderations. Furthermore, the high dimensionality of theseam usually represents far more degrees of freedom than canbe visualized straightforwardly. Traditionally, the CI seam ischaracterized by its local minima, which are termed minimalenergy CIs (MECIs).18, 19 This parallels the IRC approachfor ground state reactions in providing a convenient set ofmolecular geometries that can be used to describe a reactionmechanism. However, the ultrafast nature of photochemicalprocesses makes the relevance of these points uncertain.In fact, it has been shown in some cases that they havelittle or no role in excited state reaction mechanisms.16, 20, 21

Nevertheless, they do retain some utility for descriptivepurposes, i.e., as “signposts” on the PES that can be usedto describe and distinguish alternative mechanisms, if onlybecause there are few easily computed alternatives available.

In contrast, powerful statistical techniques exist for an-alyzing high-dimensional data sets, which could allow us tocircumvent this somewhat ad hoc approach. These so-calleddimensionality reduction techniques characterize general Rie-mannian manifolds by transformations of point clouds sam-pled from the manifold. Such methods take as input a set ofdatapoints and in return give a description of that dataset interms of a reduced set of variables. Consider, for instance,a one-dimensional manifold (i.e., a curve or trajectory) em-bedded in a high dimensional space. Ideally, a dimensional-ity reduction technique applied to points sampled from thismanifold should return a single coordinate describing motionalong the manifold whose value can be interpreted as the dis-placement from some reference point on the curve. One ofthe most common and mature of these techniques is principalcomponent analysis (PCA),22 which fits, in the least-squaressense, an N-dimensional hyperplane to a point cloud embed-ded in RN. The variances associated with each axis of the hy-perplane give a measure of the importance of each coordinate.

Typically, only a small number, M, of axes are retained, thuscreating an M-dimensional representation of the original databased only on the most strongly varying degrees of freedomand projecting out all other, more weakly varying dimensions.However, PCA’s inherent linearity strongly limits its appli-cability to large-scale chemical dynamics. In particular, theassumption of linear variance of the coordinates in the hyper-plane fitting process introduces a problematic dependence onthe choice of coordinate system when the variations becomelarge. For example, a large amplitude torsional motion may beviewed as a linear displacement in a dihedral angle or alter-natively a highly nonlinear collective motion in the Cartesiancoordinates of the atoms. In applications to chemical dynam-ics, this precludes the use of PCA for the characterization ofmany sequential reactions and the results are strongly depen-dent on the choice of coordinate system.

To address such issues, there has been increasinginterest in nonlinear dimensionality reduction (NLDR)techniques.23–26 As in PCA, these techniques quantify “im-portant” coordinates, and project out “unimportant” degreesof freedom. However, in contrast to PCA, these techniquesare often invariant to coordinate system, and can recoverparameters describing nonlinear manifolds. Rather than fo-cusing on data variance in a specified coordinate system,NLDR techniques construct low-dimensional subspaces thatpreserve short-range spatial relationships from the originalhigh-dimensional space – these are constructed by projectionsthat preserve distances between nearby datapoints in the orig-inal space, but not necessarily those between distant points.In larger, equilibrium chemical systems, NLDR techniqueshave been applied with notable successes; for instance, theISOMAP NLDR technique has been shown to be effectivein identification of reaction coordinates and transition statesdirectly from dynamics data.24 The diffusion map algorithmhas also been applied to molecular systems, allowing a reac-tion coordinate to be identified25 and its associated reducedequation of motion to be derived27 without any a priori as-sumptions about the nature of this coordinate. We also employdiffusion maps in this paper. The diffusion map techniqueis particularly well suited for ultrafast nonadiabatic chemi-cal dynamics because of its nonlinearity, coordinate indepen-dence, computational simplicity (relative to other NLDR tech-niques), and robustness to noise.28

In this paper, we specifically focus on two photoisomer-ization reactions that have already been well studied and haveknown excited state mechanisms. Our goal is to show that dif-fusion mapping recovers the known mechanisms from full di-mensional dynamics data without any guidance or foreknowl-edge.

The first example is the photoisomerization of ethyleneafter excitation to the lowest ππ* bright electronic state.Ethylene may be thought of as the paradigmatic molecule forphotoisomerization, being perhaps the simplest molecule witha C=C double bond. It has therefore long been of both ex-perimental and theoretical interest.29–37 There is a barrierlesspath from the Franck-Condon point in ethylene to a variety ofCIs connecting S1 and S0. Several geometries that have beenhighlighted in previous discussions of the photodynamics inethylene are shown in Fig. 1. (Figure 1 also shows the energet-

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22A519-3 Virshup, Chen, and Martínez J. Chem. Phys. 137, 22A519 (2012)

FIG. 1. Important points on the ππ* excited state PES of ethylene, obtainedfrom SA-3 CAS(2/2) / 6-31G** optimizations. Energy values are reportedrelative to the Franck-Condon (FC) point. Minimal energy conical intersec-tions (MECIs) are doubly underlined. Note that the level of theory employedoverestimates the Frank-Condon point energy, and that higher of levels oftheory show that the ethylidene and pyramidalized intersections should, infact, be more similar in energy.

ics that correspond to the level of theory we use in this paperfor reference below. For more accurate values of these ener-gies, we refer the reader to other publications as referencedabove.) Because all of these structures are accessible fromthe Franck-Condon point with no barrier, dynamical effectsplay a crucial role in determining the excited state reactivityof ethylene. Furthermore, the structure of the reaction pathis not easily represented linearly in standard internal coordi-nates, thus requiring ad hoc, or, as here, nonlinear definitionsof the reaction coordinates and CI seam.

The second example is a model of the chromophore inphotoactive yellow protein (PYP). This chromophore has twoisomerizable bonds in the excited state, and thus multipleexcited state reaction mechanisms may be expected. Inter-estingly, there are indications that the surrounding environ-ment can strongly influence the branching ratio between thesemechanisms.38, 39 It is also considerably larger than ethylene(21 atoms) and thus serves to show that the success of diffu-sion maps for the interpretation and analysis of excited statedynamics is not limited to small molecules.

We also use the diffusion map approach to address along-standing question in excited state dynamics around CIs– namely, the role of intersection topography in the efficiency(or lack thereof) of nonadiabatic transitions.40–42 As shownschematically in Fig. 2, the local topography around a CI maybe “peaked” or “sloped”; this can be quantified using the over-all slope introduced by Yarkony:40

S = limR0→RCI

∇(

E1(R) + E2(R)

2

)∣∣∣∣R=R0

. (1)

The length of the projection of this vector on the branch-ing plane provides a measure of the sloped character of the in-tersection – for an ideal peaked intersection, the vector S hasno projection on the branching plane. Dynamical considera-tions suggest that peaked intersections should more efficientlyfunnel trajectories towards the point of degeneracy RCI;40–42

however, a quantitative relationship between these static PESproperties and nonadiabatic dynamics has yet to be firmly es-tablished. By using diffusion maps, we are able to quantify

FIG. 2. Schematic depiction of peaked and sloped conical intersections(CIs). The character of the CI is distinguished by the norm of the gradientof the branching plane, represented by a dashed line. It has been suggestedthat peaked intersections more efficiently transfer population because theirpronounced funnel shape tends to direct trajectories toward the point of de-generacy.

the influence of intersection topography on the efficiency ofnonadiabatic transitions, demonstrating that peaked intersec-tions are indeed more likely to lead to efficient populationtransfer.

II. METHODS

A. Diffusion maps

The process of finding low-dimensional manifoldsembedded in high-dimensional spaces (NLDR) is an areaof active algorithmic research. In this article, we employdiffusion maps, an example of one such recently developedalgorithm, to quantitatively characterize the configurationspace manifold underlying excited state reaction pathwayssampled from ab initio multiple spawning (AIMS) simula-tions of the nonadiabatic dynamics. The use and constructionof diffusion maps from high-dimensional data has beendescribed extensively elsewhere27, 43–45 and we restrictourselves to a brief overview here. The basic idea is to modela diffusion process over a set of points sampled from themanifold of interest. The diffusion process elucidates theunderlying degrees of freedom by identifying local directionsalong which neighboring data points are connected, i.e.,accessible via diffusion. In doing so, a manifold containingthe entire data set can be traced out. Keeping track of the localdirections employed over the course of the diffusion results ina parameterization of the original dataset with a dimensional-ity that tends toward that of the embedded manifold and notthe larger dimension of the embedding space. It is importantto note that “diffusion mapping” is a computational, notphysical, process. Although diffusion maps have specialconnections to manifolds that arise from diffusive processes,they can be used to map geometrical manifolds of arbitraryorigin, such as the excited state dynamics described here.27

The mapping takes as input a finite collection of points{xi}Mi=1 in the original space. In this article, these points aremolecular geometries (xi ∈ R3N) sampled from AIMS tra-jectories at regular time intervals, as described below. A dif-fusion kernel is constructed on the dataset, representing theunnormalized amplitude of a random walker diffusing fromdatapoint i to point j,

kij = e−(dij /D)2, (2)

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22A519-4 Virshup, Chen, and Martínez J. Chem. Phys. 137, 22A519 (2012)

where dij is the distance between datapoints i and j, and Dis an empirically chosen length scale as described below. Adiffusion operator is then constructed from the kernel, repre-senting the probability of transition from datapoint i to j:

a(xi , xj ) = 1

Zi

kij

(vivj )α

where vi =∑

j

kij and Zi =∑

j

kij

(vivj )α, (3)

where the parameter α determines the specific type of dif-fusion modeled (discussed below), and the quantities υ i andZi are normalization and renormalization constants, respec-tively. We refer the reader to earlier in-depth discussions ofthis renormalization procedure.27, 43

An m-dimensional representation is found by solving forthe m largest eigenvalues of the diffusion operator a. Them corresponding eigenvectors {φj}j = 1, . . , m provide the low-dimensional parameterization of the original dataset. In thisrespect, the diffusion operator plays a role analogous to thatof the covariance matrix in PCA. However, while the originalhigh-dimensional coordinates are the basis of the covariancematrix, the data points themselves are the basis of the diffu-sion kernel. Thus, the dimensionality of each eigenvector isnot the dimensionality of the original coordinate system, as inPCA, but instead M, the number of data points used to con-struct the mapping. The low dimensional representation of theith datapoint, x′

i, is given by the ith components of the domi-nant m eigenvectors,

x ′i := (φ1 (i) , φ2 (i) , . . . , φm (i)) . (4)

Thus, we have constructed a mapping of the originaldataset from R3N to Rm. We refer to this low-dimensionalspace as the “reduced space,” and the m new coordinates asthe “diffusion coordinates.” The dimensionality of the param-eterization, m, is determined by examination of the eigenvaluespectrum; typically, all eigenvectors corresponding to eigen-values beneath the first significant spectral gap are discarded.

The parameter D sets the distance scale of the diffu-sion kernel (cf. Eq. (2)). This should be much larger thanthe nearest neighbor spacings between data points to smoothout discretization effects, but small enough so that large-scalefeatures of the manifold are not obscured. The choice of α

depends on the significance of data point density; α = 0 cor-responds to standard Laplacian diffusion, which is heavily in-fluenced by the local density of datapoints; α = 1 gives an ap-proximation to Laplace-Beltrami heat diffusion, resulting in adiffusion map that is invariant the density of data points; and α

= 1/2 approximates the backward Fokker-Planck operator onthe dataset, resulting in a map that emphasizes, but is not dom-inated by, the density of sampled points along the manifold.44

For this work, we employ α = 1, as it provides an approxi-mately consistent reproduction of distances across the mani-fold (that is, the magnitude of the Jacobian between diffusionspace and coordinate space depends only weakly on location).

Finally, a distance between data points needs to be de-fined (dij in Eq. (2)). Here we use the root-mean-square dis-placement between each pair of geometries, after moving thetwo geometries into maximum coincidence through rotation46

and symmetry operations (including permutation of identicalnuclei).

B. Ab initio multiple spawning dynamics

The dynamical results presented here were generatedwith the ab initio multiple spawning (AIMS) technique formolecular quantum dynamics, which has been described ex-tensively elsewhere.4, 5, 47 The AIMS wavefunction is a linearcombination of Born-Oppenheimer basis functions

�AIMS(r, R, t) =N(t)∑i=1

ci (t) ψMi(r; R) χi(R; R̃i(t), P̃i(t)),

(5)where ψJ is the electronic wavefunction on the Jth adiabaticstate, χ (R; R̃(t), P̃(t)) is a Gaussian wavepacket centered atR̃(t) with momentum P̃(t), and ci(t) are the complex ampli-tudes associated with each vibronic basis function. The pa-rameters R̃(t) and P̃(t) have an obvious classical interpreta-tion as nuclear positions and momenta, and are propagatedclassically on the PES for the electronic state with which theyare associated. Hence, the vibronic basis functions are de-noted trajectory basis functions (TBFs). At points where theBorn-Oppenheimer approximation breaks down, i.e., in thevicinity of conical intersections, population can be transferredbetween electronic states (through the evolution of the com-plex amplitudes as dictated by the solution of the Schrödingerequation in the time-evolving basis set). In order to ensurethat the appropriate basis functions are available to representthis population transfer, new TBFs are “spawned” when thecoupling between electronic states exceeds a predeterminedthreshold – it is for this reason that the upper limit N(t) in thesum of Eq. (5) is explicitly time-dependent.

The PESs for dynamics were generated on the fly us-ing ab initio quantum chemical techniques. The electronicstructure calculations of potential energy surfaces, gradi-ents, and nonadiabatic couplings were carried out usingMOLPRO.9, 48 Specifically, the calculations for ethylene usedthe state-averaged complete active space self-consistent field(SA-CASSCF) method,49, 50 with state averaging over thelowest three singlet states, an active space of two electronsin two orbitals, and the 6-31G** basis set,51 i.e., SA-3-CAS(2/2)/6-31G**. For the photoactive yellow protein (PYP)chromophore, the electronic wavefunction is described withthe SA-2-CAS(8/6)/6-31G* method. Initial conditions in bothcases are sampled from the harmonic approximation to theWigner distribution of the ground vibrational state.

III. RESULTS

A. Ethylene

The first 200 fs of ethylene photodynamics following ver-tical excitation from the ground state minimum to the ππ*state (S1 for the electronic wavefunction ansatz used here)were simulated with 600 uncoupled AIMS simulations (eachstarting with a single TBF and spawning new TBFs as dic-tated by the nonadiabatic coupling matrix elements). Due tocomputational requirements (specifically, the cost of diago-

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22A519-5 Virshup, Chen, and Martínez J. Chem. Phys. 137, 22A519 (2012)

FIG. 3. (a) Eigenvalue spectrum for diffusion map from ethylene S1 dynam-ics. There is a large spectral gap after the first nontrivial eigenvalue (φ2),indicating that the dynamics are predominantly one-dimensional. (b) Loca-tions of the sampled geometries as a function of φ2 and φ3, colored to in-dicate the time following photoexcitation. (c) Overlays of all geometries atseveral different values of φ2, illustrating the “width” of the reaction path atseveral different points. Geometries are clustered fairly tightly at each value,illustrating that φ2 is the dominant controlling variable for the dynamics.

nalizing the M × M matrix representing the diffusion kernel,where M is the number of geometries), the diffusion map wasconstructed using geometries sampled every 5 fs from the ini-tial excited state trajectory in the first 200 simulations, yield-ing a total of 6738 molecular geometries.

The distance between each pair of molecular geometrieswas computed as described above. This pairwise distancematrix was then transformed into a diffusion kernel usingEqs. (2) and (3) with α = 1 and D = 0.5 Å. This diffusion ker-nel was then diagonalized. Note that the first eigenvector (φ1)of this and any other diffusion kernel is trivial, with a uniformvalue for the entire data set – this indicates that all data pointsoccupy a single, compact cluster.27 As shown in Fig. 3(a), theeigenvalue spectrum of the diffusion kernel from the ethy-lene excited state dynamics simulations contains a large spec-tral gap after the second eigenvector. Therefore, this diffu-sion map describes a dominantly one-dimensional reactionpath parameterized by the second eigenvector φ2. In Fig. 3(b),we show the geometries (data points) in the two-dimensionalspace characterized by the second and third eigenvectors (φ2

and φ3), colored according to the time when the geometrywas accessed in the dynamics (t = 0 corresponds to the pho-toexcitation event). The coordinates of the ith geometry (datapoint) along the φ2 and φ3 axes are given by the ith elementof the second and third diffusion kernel eigenvectors, as givenin Eq. (4). Of course, restricting the diffusion map to a singlediffusion coordinate necessarily implies that there are manymolecular geometries which are described by the same valueof the diffusion coordinate. One can get some insight into thisby superimposing molecular geometries corresponding to re-stricted ranges of the diffusion coordinate. This is done inFig. 3(c), where a set of geometries from the data set gener-ating the diffusion kernel are superimposed according to their

diffusion coordinate value. Geometries with similar values ofthe diffusion coordinate can be seen to be quite similar, butwith a variance that reflects the “width” of the data set, i.e.,the degree to which further diffusion coordinates would needto be included to completely specify the geometries in the dataset.

The values of the diffusion coordinates for the geometriesin the data set as given in Eq. (4) are expressed in terms ofunitless “diffusion modes.” In order to convert these to moremeaningful real space distances, we compute the magnitudeof the Jacobian between the diffusion coordinate φ2 and thereal space coordinates R. The value of the Jacobian at φ canbe calculated by averaged finite difference,

J (φ) = ∂R∂φ2

∣∣∣∣φ

= limδ→0

⟨R − R̄φ2 − φ

⟩{φ−δ,φ+δ}

, (6)

where 〈 〉{a, b} indicates an average over all points wherea < φ2 < b, and R̄ is the average value of R in the same in-terval. The real-space distance along this coordinate can thenbe calculated as

DRC (φ2) =∫ φ2

O

|J (φ)| dφ, (7)

where O is an arbitrarily chosen reference point (here, theFranck-Condon geometry as depicted in Fig. 2).

Selected molecular geometries corresponding to motionalong this coordinate are shown in Fig. 4. The coordinateis here referred to as the “dynamical” (as opposed to in-trinsic) reaction coordinate (DRC). This one-dimensional ex-cited state reaction path is highly nonlinear in both Cartesianand internal coordinates. Motion along the DRC schemati-cally follows the excited state reaction mechanism describedabove. At low values of the dynamical reaction coordinate(DRC), the molecule is planar and similar to the Franck-Condon geometry. More positive values of the DRC corre-spond to twisting and pyramidalization about the C=C bond,followed by hydrogen atom migration across the C=C bondto form ethylidene; finally, the newly formed methyl grouptwists and bends. The time evolution of several representa-tive TBFs along the DRC is also shown. Initially, the TBFs

FIG. 4. Several representative trajectory basis functions (TBFs) along thedynamical reaction coordinate (DRC) of the S1 state of ethylene and theircorresponding geometries. The unitless diffusion coordinate φ2 (see Fig. 3),was converted to the DRC distance shown here using Eqs. (6) and (7), withthe Franck-Condon point set to a DRC value of 0 Å. TBFs may undergotorsional motion for tens of fs (DRC 0–0.5 Å) before proceeding throughpyramidalization (DRC 0.5–1.5 Å) and hydrogen migration (DRC 1.5–3 Å)to form ethylidene (DRC 3–6 Å).

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22A519-6 Virshup, Chen, and Martínez J. Chem. Phys. 137, 22A519 (2012)

oscillate for a short time between the planar and twisted ge-ometries, before falling through the pyramidalized structuresto the ethylidene minimum.

Even though we constructed the diffusion map from alimited subset of simulations, it is possible in practice to mapthe data from all 600 original simulations into the reduced di-mensionality space. The diffusion coordinates for geometriesin the 400 trajectories that were not included in the construc-tion of the diffusion map can be estimated with the Nyströmextension.52 For each new geometry y, a new diffusion ker-nel b is constructed which includes the new point as well asall the original geometries in the set S (those used to constructthe diffusion map). The reduced coordinates for the new pointare given by

y ′i = 1

λi

∑xj ∈S

b(xj , y)φi(xj ) (8)

where λi and φi(xj) are the eigenvalues and eigenvectors fromthe original diagonalization of a. This so-called “restriction”provides a transformation from configuration space R3N tothe reduced dimensional space Rm. Using this procedure, wecan carry out a reduced dimensionality analysis of the datafrom all 600 dynamical AIMS simulations, even though only200 were utilized in the construction of the diffusion map. Wenote in passing that the inverse “lifting” transformation (deter-mining the molecular geometries in the full 3N-dimensionalspace corresponding to a given point in the reduced dimen-sional space) is necessarily more complicated since informa-tion has been discarded in the reduced space. Attempts to de-fine such a transformation have so far been restricted to aninverse transformation in a stochastic sense, for example byrunning stochastic dynamics with a constraint on the reducedspace coordinates.52 In the present work, we do not requirethis “lifting” transformation and we do not consider it further.

The value of the excited state potential energy for eachsampled geometry along the DRC is shown in Fig. 5, wherethe important geometries from Fig. 2 are also shown. A re-action path similar to the one in Fig. 2 is recovered, but withmost points lying well above the excited state minima. Thisis to be expected given that the Franck-Condon point is sev-

FIG. 5. Energies of sampled points from dynamics along the reaction pathof the S1 state of ethylene.

FIG. 6. Time evolution of the reduced probability density on the excited elec-tronic state as a function of the DRC for the photodynamics of ethylene.

eral eV above the twisted/pyramidalized MECI as well as theother depicted MECIs and minima on S1.

We can also represent the nuclear wavepacket probabil-ity density as a function of the diffusion coordinate, usingthe same Monte Carlo integration that allows us to representthe time-evolving wavepacket in an arbitrary set of internalcoordinates.53 In Fig. 6, we show the time evolution of theprobability density on the excited electronic state in terms ofthe DRC. Note that as the reaction progresses, the total wave-function density drops off quickly as population quenches tothe ground state.

In order to characterize the dynamically relevant portionof the CI seam, we select the molecular geometries at which“spawning” takes place, i.e., the initial locations of the newTBFs that are added to the simulation to allow the descrip-tion of nonadiabatic events. We choose these geometries asthose with the largest magnitude of the nonadiabatic cou-pling vector connecting S0 and S1 during the placement ofthe newly spawned TBF. Population transfer to these newlyspawned TBFs is evaluated by monitoring the complex am-plitudes c(t) as given in Eq. (5) until the associated popula-tion (evaluated as a Löwdin-like population54 to account fornonorthogonality) of the TBF becomes approximately con-stant, typically within twenty femtoseconds of the introduc-tion of the new TBF. In Fig. 7, the population transferred toground state TBFs is shown as a function of both time and thelocation of the associated spawning geometries in the reducedspace. The spawning geometries were projected in the re-duced space using the Nyström extension as discussed above.Population transfer is localized to geometries in two regionsof the DRC. As can be determined from Fig. 4, the firstof these (0.5–1.5 Å) corresponds to twisted/pyramidalizedgeometries and the second (3–6 Å) region corresponds toethylidene-like geometries where a hydrogen atom has mi-grated to form CH3CH. Furthermore, there is a strong tem-poral/spatial correlation with early population transfer occur-ring in the twisted/pyramidalized region and later populationtransfer events being mediated by ethylidene-like geometries.

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22A519-7 Virshup, Chen, and Martínez J. Chem. Phys. 137, 22A519 (2012)

FIG. 7. Population transferred to ground state as a function of DRC and time.Population is initially quenched in the pyramidalized region, while the longertimescale decay takes place in the ethylidene region.

The efficiency of population decay through nonadiabatictransitions is not constant along the reaction path and it wouldbe interesting to correlate the transition rates with features ofthe potential energy surface such as the energy or topogra-phy of the closest conical intersection. We evaluate a time-averaged rate constant for ground state quenching at eachvalue along the excited state DRC directly from the simula-tion data. Specifically, we compare the amount of populationtransfer through each point to the total wavepacket probabilitydensity that passed through that region of the path

α(x) = T (x)∫ tfti

dtρ(x, t), (9)

where x is the DRC, T(x) is the amount of population trans-ferred through that point (integral over the time coordinatein Fig. 7), and ρ(x, t) is the probability density shown inFig. 6. The DRC-dependent nonadiabatic transition rate α(x)was evaluated by a histogramming procedure, integrating overthe time coordinate in Figs. 6 and 7. The upper panel of Fig. 8shows the results, where the red bars denote the amount ofpopulation transferred to the ground state as a function of theDRC and the blue bars denote the time-integrated probability

FIG. 8. Upper panel: Population transferred (red) and time-integratedwavepacket probability density (blue) along the diffusion reaction coordinateof the S1 state of ethylene. Lower panel: Average slope of conical intersec-tions closest to the spawning points (green, left axis) and effective populationdecay rate (red, right axis). Although the pyramidalized/hydrogen geome-tries are higher in energy than ethylidene (with the electronic structure ansatzused in this work), they nevertheless serve to promote nonadiabatic transi-tions more effectively.

FIG. 9. Relationship between the rate of population transfer induced bynonadiabatic transitions and the slope of the nearest conical intersection (ob-tained from the data in Fig. 8) in ethylene. Error bars indicate the width ofthe distribution of slope parameters for intersections corresponding to thedata point.

of finding the nuclei in a geometry corresponding to a partic-ular value of the DRC.

We then calculated the average slope of conical intersec-tions as a function of the DRC by calculating the magnitudeof the overall slope vector S (Eq. (1)) for each spawning ge-ometry. The average of this distribution for each value of theDRC is shown as the green line in the lower panel of Fig. 8,showing that intersections in the twisted/pyramidalized regionof the DRC tend to be quite peaked (small overall slope) andthose in the ethylidene region of the DRC tend to be stronglysloped. This panel also shows the nonadiabatic transition rate(cf. Eq. (9)) as a red line, demonstrating that nonadiabatictransitions are far more efficient in the region of the DRCcorresponding to twisted/pyramidalized geometries, as com-pared to the ethylidene geometries. From this data, it is easyto extract the rate of nonadiabatic transitions as a function ofconical intersection slope. This is shown by plotting the de-cay rate as a function of the intersection slope in Fig. 9. Theerror bars along the x axis correspond to the width of the dis-tribution of intersection slopes for each value of the DRC.Of course, there is only limited data and this is for a spe-cific example, but Fig. 9 does show that highly sloped inter-sections tend to lead to less efficient nonadiabatic transitions,while the most efficient nonadiabatic transitions occur aroundpeaked intersections. Although there is no clear one-to-one re-lationship between intersection slope and nonadiabatic transi-tion rate (and neither is this to be expected since the momen-tum of the nuclei should also play a role), the maximal val-ues along the plot suggest a limiting relationship, supportingprevious suggestions40–42 about the importance of intersectiontopography.

B. Photoactive yellow protein chromophore

As a second example, showing that diffusion mappingalso successfully recovers low-dimensional manifolds of thedynamics of larger systems, we turn to the case of photoi-somerization in the chromophore of photoactive yellow pro-tein (PYP). We choose a model chromophore here, anionicp-coumaric acid (pCK-), shown in Fig. 10. We have pre-viously discussed the photodynamics of this chromophorein isolation and in condensed phase environments.10, 38, 55

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22A519-8 Virshup, Chen, and Martínez J. Chem. Phys. 137, 22A519 (2012)

FIG. 10. The p-coumaric acid anion, a model for the chromophore of pho-toactive yellow protein (PYP), denoted as pCK- in the text. After photoex-citation, there are two possible photoisomerization pathways – torsion aboutthe phenyl-adjacent single bond (φS) or the side chain double bond (φD),which are indicated in the figure.

Upon photoabsorption in the protein environment, the PYPchromophore undergoes trans-cis isomerization, initiating thenegative photoaxis pathway of Halorhodospira halophile.56

Various experimental57, 58 and theoretical55, 59–64 studies haveshown rotation of the ethylenic double bond (D) and thephenyl-adjacent single bond (S) are the principal reaction co-ordinates and these can compete with each other.

Here, we simulate the excited state dynamics of isolatedpCK- with 16 initial AIMS trajectories. Electronic structurecalculations were at the SA2-CAS(8/6) level of theory withthe 6-31G* basis set. In contrast to ethylene, the larger sizeand longer excited state lifetime of pCK- limits the amount ofdynamical data available. Even with a relatively small dataset,however, the diffusion mapping recovers a one-dimensionalrepresentation of the reaction path that agrees with the quali-tative picture of the reaction proceeding through torsion abouteither the S or D bond (these torsions are indicated in Fig. 10).In contrast to ethylene, where the excited state reaction dy-namics largely follows a sequential path from the Franck-Condon region to twisted/pyramidalized and finally ethyli-dene geometries, two potential outcomes compete in the pho-todynamics of pCK-. Specifically, the molecule starts in theFranck-Condon region, and then may isomerize about eitherthe S or the D bond. As shown in Fig. 11, the dynamical re-action coordinate revealed by diffusion mapping captures thisbehavior nicely. Negative values of the DRC correspond totorsion about the D bond, and positive values correspond totorsion about the S bond. Geometries at the extremes (bothpositive and negative) of the DRC involve some degree of tor-

FIG. 11. Plot of the two relevant torsion angles along the DRC for the modelPYP chromophore. The diffusion map correctly identifies these as the twomost important coordinates in the photodynamics of the chromophore, asshown by the fact that the DRC is dominated by the φS torsion angle forthe right half of the plot (from 0 to 9 Å) and by the φD torsion angle for theleft half of the plot (from 0 to −7.5 Å).

FIG. 12. Time evolution of the TBFs along the DRC of the model PYPchromophore. The majority undergo torsion about the phenyl-adjacent sin-gle bond (denoted by positive values of the DRC). Points where new TBFsare spawned are indicated by diamonds, and comparison with Fig. 11 showsthat these correspond to a φD torsion angle around 90◦.

sion about both bonds. Of course, a more complete picturewould be obtained by including the second eigenvector of thediffusion operator to build a two-dimensional reduced spacepicture of the excited state dynamics.

As shown in Fig. 12, most of the TBFs undergo torsionaround the S bond after photoexcitation and continue to os-cillate about φS = 90◦ (DRC = 5 Å) for at least 1 ps. How-ever, no spawning events occur in this region of the reactionpath. Instead, spawning events occur only in regions wherethe molecule is twisted about the D bond. This can occur intwo ways; either by immediate twisting about the D bond orby torsion about the S bond followed by torsion about the Dbond (so that both bonds are twisted in a “hula-twist” likemotion).65

IV. CONCLUSIONS

Our results strongly suggest that diffusion maps providea useful low-dimensional description of ultrafast nonadiabaticdynamics on excited state manifolds, as demonstrated in thetwo reactions studied. For the ππ* excited state dynamicsof ethylene, the reduced dimensional model produced a one-coordinate representation that agreed with previous dynami-cal studies. The dynamical reaction coordinate derived fromthe diffusion map has a natural interpretation of a highlynonlinear interpolation between torsional rotation, pyrami-dalization, and hydrogen migration coordinates, without anya priori choice of coordinate parameterization. This nonlin-earity along an unusual set of coordinates underlines the needfor nonlinear analyses of such processes, as linear methodssuch as PCA will only yield useful results if a “good” set ofreaction coordinates is known in advance.

We find an inverse correlation between overall CI slopeand the rate of population transfer, as has been suggested inprevious work. The peaked (unsloped) CI transfers popula-tion more efficiently, despite being higher in energy than thesloped CI. While suggestive of an overall trend, our resultsalso show a need to disentangle aspects of dynamics on theoverall PES from those specifically related to the CI topog-raphy, so as to further elucidate the detailed relationship be-tween CI topography and nonadiabatic transition rates.

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22A519-9 Virshup, Chen, and Martínez J. Chem. Phys. 137, 22A519 (2012)

Diffusion map analysis also identified a one-dimensionalmanifold in the excited state dynamics of anionic p-coumaricacid, a model system for the chromophore of photoactive yel-low protein. The dynamical reaction coordinate picks out bothof the two competing torsional relaxations via the ethylenicdouble bond (D) and the phenyl-adjacent single bond (S), andthus allowed the classification of trajectories into two cate-gories based on which torsional relaxation pathway was cho-sen. The data furthermore showed a clear trend of triggeringspawning events only when there is significant torsion aboutthe D bond.

The approach in this work can be generalized in a numberof different ways. The distance metrics presented here werebased only on the 3N-dimensional molecular conformationand were restricted to a single electronic surface. A distancemetric based on 6N-dimensional phase space would be use-ful in further disentangling the dynamical and static aspectsof population transfer near conical intersections. Addition-ally, an extended distance metric that included both nuclearconfigurations and electronic information – for instance, al-lowing nonadiabatic mixing to bring trajectories on differentelectronic states close to one another – could provide a pictureof dynamical time-evolution across multiple electronic states.The NLDR techniques presented here can also be straight-forwardly applied to other semi-classical models of chem-ical dynamics, such as surface hopping66 and wavepacketdynamics,67 and a diffusion map of a dynamical quantum sys-tem could easily be constructed on a collection of points sam-pled from the probability density corresponding to the nuclearwavefunction.

We have demonstrated that nonlinear dimensionality re-duction techniques such as diffusion maps are powerful newstatistical tools for analyzing dynamics simulation data. Dy-namically relevant reaction coordinates can be deduced au-tomatically without the need for a priori specification of in-teresting coordinates. Furthermore, these reaction coordinateshave illuminated qualitative features of the dynamics them-selves, such as the nature and location of spawning pointsin ab initio multiple spawning simulations, the branching ra-tios of different reaction channels, and the dependence ofnonadiabatic transitions on the slope of CIs. The dynami-cal reaction coordinates remain well defined even far fromequilibrium, and thus afford a new and valuable tool forunderstanding ultrafast processes on electronically excitedstates.

Although we have restricted our use of diffusion mapsto analysis of data from dynamics carried out in full dimen-sionality, one can also imagine using the resulting diffusionmap to construct a reduced dimensionality model that mightbe used in its own right. For example, an explicit reduced di-mensionality model could provide a means of reaching longertime scales (as has been explored for ground state reactions27)or calculating the wavepacket evolution with more nearlynumerically exact quantum dynamics methods.68 This is anavenue which might be profitable in future work, but thenonequilibrium character of excited state reactions may in-troduce difficulties in determining how to treat the degrees offreedom outside the set of coordinates selected by the diffu-sion map.

ACKNOWLEDGMENTS

This work was supported by the National Science Foun-dation through Grant No. CHE-11-24515 with computationalsupport through DOE (Contract No. DE-AC02-7600515). Wethank Professor Dr. Dirk Hundertmark (Karlsruher Institut fürTechnologie) for helpful discussions.

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